Guidance

Mathematics Written Examination Syllabus

Published 6 May 2014

1. Arithmetic

1.1 Expresses quantities in the form of a ratio, proportion or percentage

1. Compares two quantities of the same kind by expressing one as a Ratio of the other.
2. States that proportion is an equation of ratios.
3. States that percentage is a ratio multiplied by 100.
4. Expresses fractional and decimal quantities in the form of a percentage.
5. Expresses an increase or gain as a percentage.
6. Expresses a decrease or contraction as a percentage.
7. Expresses an error as a percentage.
8. Solves problems related to 1.1.1 to 1.1.7.
9. Understands similarity and proportion; simple objects to scale (length, area, volume and mass).
10. Understands rates, averages, proportional rates of doing work and cost.
11. Understands concepts such as “man hours”, “kWh”, etc.
12. Solves problems related to 1.1.9 to 1.1.11.

2. Algebra

2.1 Uses the rules of Algebra and solves associated problems

1. Represents quantities by numbers, letters and symbols.
2. Adds algebraic quantities, both positive and negative.
3. Subtracts algebraic quantities, both positive and negative.
4. States the effect of plus or minus signs in front of a bracketed quantity or quantities.
5. States the effect of the plus or minus signs in the multiplication and division of quantities.
6. Defines the term index (power).
7. States what is meant by fractional, negative and zero indices.
8. States the rules for addition, subtraction and product of indices.
9. Solves problems related to 2.1.6, 2.1.7 and 2.1.8.
10. States the ‘Law of Distribution’.
11. States the product of two binomial expressions.
12. States the square of a binomial expression (a ± b)2.
13. States the product of the sum and difference of two algebraic quantities (a + b) (a - b).
14. Expands (a ± b)3 and factors of a3 + b3.
15. Solves problems involving the multiplication and division of polynomial expressions by binomial expressions.
16. Factorises expressions which have one factor consisting of one term only.
17. Factorises expressions of four terms which can be expressed as the product of two binomials.
18. Factorises expressions of the type ax2 + bx + c, where a, b and c have numerical values, including both:
    1. cases when a is equal to 1;
    2. cases when a is not equal to 1.
19. Factorises trinomials which form a perfect square.
20. Factorises the difference of two squares.
21. Solves problems involving the addition and subtraction of algebraic fractions.
22. Solves problems involving the multiplication and division of algebraic fractions (both 2.1.21 and 2.1.22 to be limited to polynomials no greater than binomial expressions).
23. Defines an equation as a statement of equality.
24. Simplifies and solves linear equations.
25. Understands the axioms:
    1. if equal quantities be added to two quantities that are already equal, the results will be equal;
    2. if equal quantities be subtracted from two quantities that are already equal, the remainders will be equal;
    3. equal quantities when multiplied or divided by the same quantity will give results that are equal.
26. Solves problems on the transposition of algebraic expressions.
27. Develops linear equations consistent with data provided in a question, and finds the solution to these equations.
28. Solves linear simultaneous equations of two unknowns:
    1. by the method of substitution;
    2. by the method of elimination.
29. Solves linear simultaneous equations of three unknowns.
30. Develops linear simultaneous equations of two unknowns consistent with data provided in a question, and finds the solution to those equations.
31. States what is meant by the roots of a quadratic equation.
32. Solves quadratic equations that factorise.
33. States the general formula for solution of a quadratic ax² + bx + c = 0.
34. Solves quadratic equations using the general formula.
35. Solves simultaneous equations of two unknowns consisting of linear and quadratic equations.
36. Describes direct and inverse variation.
37. Describes the use of the constant of variation.
38. Solves problems involving 2.1.35 and 2.1.36.

3. Logarithms

3.1 Uses logarithms to under take simple calculations (not directly examinable but such knowledge will be assumed)

1. Defines logarithms.
2. States laws of logarithms.
3. Uses laws of logarithms to evaluate powers etc.
4. States base of natural logarithms.
5. Evaluates expressions involving natural logarithms.

4. Graphs

4.1 Discusses the graphic representations of numerical quantities

1. States that graph axis are abscissa and ordinate, and indicates their positions.
2. Defines the dependent and independent variables.
3. Identifies the axis on which the dependent and independent variables are plotted.
4. Determines plotting points, having been given or having calculated x and y values.
5. Determines suitable scales for plotting values calculated at 4.1.4.
6. Plots linear and non-linear graphs (scales to be given in examination).
7. States that for a linear graph, only two plotting points are required.
8. States that plotting points may be given in the form: x = 1, y = 2, or (1,2).
9. States that the law of a straight line graph is of the form: y = ax + b, and defines a and b.
10. Writes the equation y = aX² + b in the form of a straight line.
11. Solves graphically problems of the form pVⁿ = C, where n is unknown.
12. States that two simultaneous equations plotted as graphs on the same axis have solutions where the graphs intersect.
13. States that the solution to a quadratic equation is given by the points where the graph of the quadratic equation crosses the x-axis, i.e. where y = 0.
14. States that the solution to simultaneous quadratic equations is given by the points where the graphs of the equations intersect.
15. Solves equations by graphical addition.
16. Solves graphic problems of trigometric form no more complex than y = a sin mx + b cos nx, and finds the solution of simultaneous equations involving such graphs.
17. Solves graphical problems of the form y = a tan mx.

5. Trigonometry

5.1 Discusses and uses the basic laws of trigonometry

1. States that angles are measured in degrees or radians and relates the two.
2. Defines acute, right, obtuse and reflex angles.
3. Defines complementary angles and supplementary angles.
4. Defines Sine, Cosine, Tangent, Secant, Cosecant, Cotangent and the relationships between them.
5. Determines Sin, Cos and Tan from given right angled triangle.
6. Reads values of Sin, Cos, Tan, Sec, Cosec and Cot for any angle between 0’ and 90’.
7. Determines an angle from tables knowing its sin, cos, tan, sec, cosec or cot.
8. Determines values of sin, etc, for angles 90’ - 360’ and also is able to obtain an angle (00 - 360’) knowing its sin, etc.
9. States the theorem of Pythagoras.
10. Solves right angled triangles for any side or angle.
11. States the Sine Rule.
12. States the Cosine Rule.
13. Solves any triangle for any side or angle using the above rules.

6. Mensuration

6.1 Solves problems related to plane figures and solids

1. States the formulae for the determination of the areas of a rectangle, parallelogram, triangle, polygon, trapezium, circle, annulus, ellipse, segment and sector.
2. Determines the area of a triangle, given:
   1. all three sides;
   2. two sides and an included angle;
   3. the base and vertical height.
3. Solves problems involving 6.1.1 and 6.1.2 to include the application of trigonometry and geometry as specified in previous objectives.
4. Determines the mean height of a figure from area and length.
5. States the formulae for determining the volume of a cube, oblong, cylinder, cone, square, pyramid and sphere.
6. Determines masses of solids at 6.1.5.
7. Determines the surface area of solids given at 6.1.5 (formulae for sphere to be given).

7. Calculus - differentiation

7.1 Discusses differential calculus and solves associated problems

1. Determines the gradient of a chord.
2. Discusses the concept of elemental lengths x and y.
3. Discusses the meaning of the limiting value of δy/δx as X - 0, defining it as dy/dx.
4. Derives the derivative of axⁿ where n is +ve or -ve.
5. Determines the derivatives of multinomial algebraic expressions.
6. States the derivative of a constant.
7. Discusses the concept of 2nd derivatives.
8. Repeats 7.1.5 for 2nd derivatives.
9. States the derivatives for sinx, cosx and lnx.
10. Determines the 1st derivatives of functions involving.
11. Discusses the concept of rate of change.
12. Determines velocity from displacement-time functions and acceleration from velocity-time functions.
13. States that at the turning point of a curve, the differential coefficient is zero.
14. Discusses the concept of maximum and minimum.
15. Identifies max/min values for examination of 2nd derivative.
16. Determines the max and/or min volumes for given functions.
17. Writes derivatives in terms of functional notation.

8. Calculus - integration

8.1 Discusses integral calculus and solves associated problems

1. States that integration is the reverse of differentiation.
2. Discusses the concept of the indefinite integral and the need for a constant.
3. States the integral of axⁿ where n 3≠ -1.
4. Determines the integrals of multinomial algebraic expressions by applying 8.1.3.
5. Determines the constant of integration from given conditions.
6. Discusses the concept of limits.
7. Repeats 8.1.4 and includes limits.
8. States the integrals of sinx and cosx.
9. Determines the integrals of functions involving 8.1.8.
10. Discusses the concept of elemental summation to determine areas and volumes and relates this to integration.
11. Determines areas and volumes by integration given the law of the boundary curve and limits.
12. Derives expressions for the area under the curve, given by pVⁿ = C.
13. Solves problems relating to 8.1.12.